L(s) = 1 | − 3-s − 5·5-s + 3·7-s − 9-s + 3·11-s + 4·13-s + 5·15-s + 6·17-s − 3·19-s − 3·21-s + 2·23-s + 7·25-s − 7·27-s − 5·29-s − 4·31-s − 3·33-s − 15·35-s − 7·37-s − 4·39-s − 7·41-s − 43-s + 5·45-s + 11·47-s + 6·49-s − 6·51-s − 3·53-s − 15·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2.23·5-s + 1.13·7-s − 1/3·9-s + 0.904·11-s + 1.10·13-s + 1.29·15-s + 1.45·17-s − 0.688·19-s − 0.654·21-s + 0.417·23-s + 7/5·25-s − 1.34·27-s − 0.928·29-s − 0.718·31-s − 0.522·33-s − 2.53·35-s − 1.15·37-s − 0.640·39-s − 1.09·41-s − 0.152·43-s + 0.745·45-s + 1.60·47-s + 6/7·49-s − 0.840·51-s − 0.412·53-s − 2.02·55-s + ⋯ |
Λ(s)=(=((218⋅73⋅193)s/2ΓC(s)3L(s)−Λ(2−s)
Λ(s)=(=((218⋅73⋅193)s/2ΓC(s+1/2)3L(s)−Λ(1−s)
Degree: |
6 |
Conductor: |
218⋅73⋅193
|
Sign: |
−1
|
Analytic conductor: |
313997. |
Root analytic conductor: |
8.24431 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
3
|
Selberg data: |
(6, 218⋅73⋅193, ( :1/2,1/2,1/2), −1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 7 | C1 | (1−T)3 |
| 19 | C1 | (1+T)3 |
good | 3 | S4×C2 | 1+T+2T2+10T3+2pT4+p2T5+p3T6 |
| 5 | S4×C2 | 1+pT+18T2+48T3+18pT4+p3T5+p3T6 |
| 11 | S4×C2 | 1−3T+8T2+10T3+8pT4−3p2T5+p3T6 |
| 13 | S4×C2 | 1−4T+23T2−96T3+23pT4−4p2T5+p3T6 |
| 17 | S4×C2 | 1−6T+11T2+20T3+11pT4−6p2T5+p3T6 |
| 23 | S4×C2 | 1−2T+49T2−60T3+49pT4−2p2T5+p3T6 |
| 29 | S4×C2 | 1+5T+60T2+252T3+60pT4+5p2T5+p3T6 |
| 31 | S4×C2 | 1+4T+49T2+184T3+49pT4+4p2T5+p3T6 |
| 37 | S4×C2 | 1+7T+92T2+432T3+92pT4+7p2T5+p3T6 |
| 41 | S4×C2 | 1+7T+134T2+572T3+134pT4+7p2T5+p3T6 |
| 43 | S4×C2 | 1+T+94T2+58T3+94pT4+p2T5+p3T6 |
| 47 | S4×C2 | 1−11T+174T2−1050T3+174pT4−11p2T5+p3T6 |
| 53 | S4×C2 | 1+3T+96T2+80T3+96pT4+3p2T5+p3T6 |
| 59 | S4×C2 | 1+3T+132T2+246T3+132pT4+3p2T5+p3T6 |
| 61 | S4×C2 | 1+7T+170T2+852T3+170pT4+7p2T5+p3T6 |
| 67 | S4×C2 | 1−12T+197T2−1592T3+197pT4−12p2T5+p3T6 |
| 71 | S4×C2 | 1+9T+212T2+1270T3+212pT4+9p2T5+p3T6 |
| 73 | S4×C2 | 1+107T2−392T3+107pT4+p3T6 |
| 79 | S4×C2 | 1+15T+278T2+2386T3+278pT4+15p2T5+p3T6 |
| 83 | S4×C2 | 1+16T+3pT2+2208T3+3p2T4+16p2T5+p3T6 |
| 89 | S4×C2 | 1+3T+242T2+556T3+242pT4+3p2T5+p3T6 |
| 97 | S4×C2 | 1+5T+270T2+872T3+270pT4+5p2T5+p3T6 |
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L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.11112182115085410648606980086, −6.95771083105017229074348701386, −6.91134600153018472074665403871, −6.82815965780422193161091147174, −6.10288505403354802736893225828, −5.96052398828219944581829884458, −5.87926827515724052261598016820, −5.52001643745958183784302632548, −5.35977906315062855585023326234, −5.35371292017920075780715973889, −4.75295064691961968010898748900, −4.48925849427397303278459854767, −4.41228469564613108722655616541, −4.00095373329848451302148825250, −3.92025522831354794269075256813, −3.68222479185258329361392215653, −3.49561870694169322802113584692, −3.29812113484374428356532447535, −2.95908749824610114709844453550, −2.43984940371804617205461236204, −2.24055631838275322536401146455, −1.64258285855059893799413357882, −1.57283399695533435773204712778, −1.14871470525869076832307503918, −1.04303462841050887097293724808, 0, 0, 0,
1.04303462841050887097293724808, 1.14871470525869076832307503918, 1.57283399695533435773204712778, 1.64258285855059893799413357882, 2.24055631838275322536401146455, 2.43984940371804617205461236204, 2.95908749824610114709844453550, 3.29812113484374428356532447535, 3.49561870694169322802113584692, 3.68222479185258329361392215653, 3.92025522831354794269075256813, 4.00095373329848451302148825250, 4.41228469564613108722655616541, 4.48925849427397303278459854767, 4.75295064691961968010898748900, 5.35371292017920075780715973889, 5.35977906315062855585023326234, 5.52001643745958183784302632548, 5.87926827515724052261598016820, 5.96052398828219944581829884458, 6.10288505403354802736893225828, 6.82815965780422193161091147174, 6.91134600153018472074665403871, 6.95771083105017229074348701386, 7.11112182115085410648606980086